Finitely Presented Semigroups

• Introduction
• The Construction of Free Semigroups and their Elements
  • Structure Constructors
    • FreeSemigroup(n): RngIntElt → SgpFP
    • FreeMonoid(n): RngIntElt → MonFP
    • Example: Free Semigroup
  • Element Constructors
    • S ! [i₁, ... iₛ]: SgpFP, [RngIntElt] → SgpFPElt
    • Id(M): MonFP → MonFPElt
    • M ! 1: MonFP → MonFPElt
• Elementary Operators for Words
  • Multiplication and Exponentiation
    • u * v: SgpFPElt, SgpFPElt → SgpFPElt
    • u ^ n: SgpFPElt, RngIntElt → SgpFPElt
    • G ! Q: SgpFP, [  SgpFPElt ] → SgpFPElt
  • The Length of a Word
    • # u: SgpFPElt → RngIntElt
  • Equality and Comparison
    • u eq v: SgpFPElt, SgpFPElt → BoolElt
    • u ne v: SgpFPElt, SgpFPElt → BoolElt
    • u lt v: SgpFPElt, SgpFPElt → BoolElt
    • u le v: SgpFPElt, SgpFPElt → BoolElt
    • u ge v: SgpFPElt, SgpFPElt → BoolElt
    • u gt v: SgpFPElt, SgpFPElt → BoolElt
    • IsOne(u): MonFPElt → BoolElt
• Specification of a Presentation
  • Relations
    • w₁ = w₂: SgpFPElt, SgpFPElt → Rel
    • LHS(r): Rel → SgpFPElt
    • RHS(r): Rel → SgpFPElt
  • Presentations
    • Semigroup< generators | relations >: SgpFPElt, ..., SgpFPElt, Rel, ...Rel → SgpFP
    • Monoid< generators | relations >: MonFPElt, ..., MonFPElt, Rel, ..., Rel → MonFP
    • Example: Monoid
  • Accessing the Defining Generators and Relations
    • S . i: SgpFP, RngIntElt → SgpFPElt
    • Generators(S): SgpFP → { SgpFPElt}
    • NumberOfGenerators(S): SgpFP → RngIntElt
    • Ngens(S): SgpFP → RngIntElt
    • Parent(u): SgpFPElt → SgpFP
    • Relations(S): SgpFP → [ Rel ]
• Subsemigroups, Ideals and Quotients
  • Subsemigroups and Ideals
    • sub<S | L₁, ..., Lᵣ>: SgpFP, SgpFPElt, ..., SgpFPElt → SgpFP
    • ideal<S | L₁, ..., Lᵣ>: SgpFP, SgpFPElt, ..., SgpFPElt → SgpFPIdl
    • lideal<G | L₁, ..., Lᵣ>: SgpFP, SgpFPElt, ..., SgpFPElt → SgpFPIdl
    • rideal<G | L₁, ..., Lᵣ>: SgpFP, SgpFPElt, ..., SgpFPElt → SgpFPIdl
  • Quotients
    • quo< F | relations >: SgpFP, Rel, ..., Rel → SgpFP
• Extensions
  • DirectProduct(R, S): SgpFP, SgpFP → SgpFP
  • FreeProduct(R, S): SgpFP, SgpFP → SgpFP
• Elementary Tietze Transformations
  • AddRelation(S, r): SgpFP, Rel → SgpFP
  • AddRelation(S, r, i): SgpFP, Rel, RngIntElt → SgpFP
  • DeleteRelation(S, r): SgpFP, Rel → SgpFP
  • DeleteRelation(S, i): SgpFP, RngIntElt → SgpFP
  • ReplaceRelation(S, r₁, r₂): SgpFP, Rel, Rel → SgpFP
  • ReplaceRelation(S, i, r): SgpFP, RngIntElt, Rel → SgpFP
  • AddGenerator(S): SgpFP → SgpFP
  • AddGenerator(S, w): SgpFP, SgpFPElt → SgpFP
  • DeleteGenerator(S, y): SgpFP, SgpFPElt → SgpFP
• String Operations on Words
  • Eliminate(u, x, v): SgpFPElt, SgpFPElt, SgpFPElt → SgpFPElt
  • Match(u, v, f): SgpFPElt, SgpFPElt, RngIntElt → BoolElt, RngIntElt
  • Random(S, m, n): SgpFP, RngIntElt, RngIntElt → SgpFPElt
  • RotateWord(u, n): SgpFPElt, RngIntElt → SgpFPElt
  • Substitute(u, f, n, v): SgpFPElt, RngIntElt, SgpFPElt, RngIntElt → SgpFPElt
  • Subword(u, f, n): SgpFPElt, RngIntElt, RngIntElt → SgpFPElt
  • ElementToSequence(u): SgpFPElt → [ SgpFPElt ]
  • Eltseq(u): SgpFPElt → [ SgpFPElt ]